Final answer:
To find dy/dx by implicit differentiation of the equation 4x^4+7x^2y−6xy^4 = 4, differentiate each term with respect to x, collect all dy/dx terms on one side, and solve for dy/dx.
Step-by-step explanation:
The question asks us to find dy/dx by implicit differentiation of the equation 4x4+7x2y-6xy4 = 4. To do this, we differentiate both sides of the equation with respect to x, while treating y as a function of x (y = y(x)).
Start by differentiating each term separately:
- The derivative of 4x4 with respect to x is 16x3.
- The derivative of 7x2y is 14xy + 7x2(dy/dx) since it's a product of functions (using the product rule and chain rule).
- The derivative of -6xy4 is -6y4 - 24x(y3)(dy/dx) (also using the product rule and chain rule).
- The derivative of the constant 4 is 0.
After differentiating, we obtain the equation:
16x3 + (14xy + 7x2(dy/dx)) - (6y4 + 24x(y3)(dy/dx)) = 0.
We then solve for dy/dx by collecting all the dy/dx terms on one side and the remaining terms on the other side, and factoring out dy/dx.